Determining subspaces of $P_3$ 
Theorem: If $W$ is a set of one or more vectors in a vector space $V$, then $W$ is a subspace of $V$ if and only if the following conditions are satisfied. 



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*If $\mathbf{u}$ and $\mathbf{v}$ are vectors in $W$, then $\mathbf{u+v}$ is in $W$.

*If $k$ is a scalar and $\mathbf{u}$ is a vector in $W$, then $k \mathbf{u}$ is in $W$.


Use the theorem to determine which of the following are subspaces of $P_3$.


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*All polynomials $a_0 +a_1 x + a_x x^2 +a_3x^3$ for which $a_0=0$.

*All polynomials $a_0 +a_1 x + a_x x^2 +a_3x^3$ for which $a_0+a_1+a_2+a_3=0$.

*All polynomials of the form $a_0 +a_1 x + a_x x^2 +a_3x^3$ in which $a_0,a_1,a_2,$ and $a_3$ are rational numbers.

*All polynomials of the form $a_0+a_1x$, where $a_0$ and $a_1$ are real numbers.


I'm having trouble understanding how this theorem relates to these generalized examples here..  
Such as (a), I'm assuming $a_0, a_1$, etc are vectors like $u$ and $v$, and $P_3$ is $V$, but I'm not sure what $W$ is precisely, is it all polynomials that fit that description? I'm kind of forgetting how to show if the form is $P_3$ mathematically, but I'm assuming it is because it's rank $3$. But what does $a_0 = 0$ have to do with determining it? 
How am I supposed to test if "all polynomials" are in $W$? Is anyone else good at these generalized vector space problems? I'm having trouble getting a good grasp of real vector spaces and subspaces for answering relative problems...
 A: Let's try to breakdown understanding in several steps for clarification.


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*$P_3$ is a vector space, the vector space of all polynomials of degree 3 or less. As elements of a vector spaces are called vectors this means all polynomials of degree 3 or less are vectors in this space. Hence, we see that $x^3$ is a vector, $x^2-3$ is a vector, $5x$ isa vector, the combination of them $x^3 +x^2 +5x -3$ is a vector, etc.

*The coefficients you have referred to are not vectors. They are what they are, coefficients. There is one subtlety: the only case where they coincide is for $a_0$. This means in the preceding example $-3$ is a vector and a coefficient. Two structures at the same time.

*Before any precise notions of "rank" have come forth we should avoid this usage. It's not going to shed light on the discussion.

*Subspaces are subsets of a vector space which you hope to preserve structure. This is useful because sometimes you can obtain information about a space just by noticing it is part of a greater structure and inherits that structure, empowering your understanding. The examples put forth by your exercise are strict (possible) subspaces. Notice that by the definition of subspaces you have that the space itself is a subspace and that the set $\{ 0 \}$ is also a subspace. These are called the trivial subspaces.

*To verify that these subsets of $P_3$ are indeed subspaces we need to check those conditions. They are saying essentially that if we perform addition of elements with those characteristics and multiplication by numbers they preserve those characteristics. 


With this in mind, let's determine if $$W_1 = \{ p \in P_3 \, : \, p(x) = a_0+a_1x + a_2x^2 +a_3x^3, \quad a_0 = 0 \}$$ is a subspace of $P_3$. Since $a_0=0$ then really all polynomials in $W_1$ are of the form $$p_1(x) = a_1x + a_2x^2 +a_3x^3.$$ Now we need to check the conditions. If we sum another polynomial $p_2 (x) = b_1 x +b_2x^2 +b_3x^3$ to $p_1(x)$, is the sum still a polynomial with null independent coefficient? If we multiply either of those by a number, will the independent coefficient still be zero? We have $$p_1(x) + p_2(x) = (a_1+b_1)x + (a_2+b_2)x^2 +(a_3+b_3)x^3 = c_1x +c_2 x^2 +c_3x^3,$$ therefore the sum is still a polynomial with zero independent coefficient. Multiplying by a number gives $$k p_1(x) = (ka_1) x + (ka_2)x^2 +(ka_3)x^3 = d_1 x + d_2x^2 +d_3x^3,$$ and it's still of the form given. Since the conditions are satisfied, we've proved $W_1$ is a subspace of $P_3$. Check the others.
